Nearly Kähler manifold

In mathematics, a nearly Kähler manifold is an almost Hermitian manifold $M$ , with almost complex structure $J$ , such that the (2,1)-tensor $\nabla J$ is skew-symmetric. So,

$(\nabla_X J)X =0 \,$

for every vectorfield $X$ on $M$ .

In particular, a Kähler manifold is nearly Kähler. The converse is not true. The nearly Kähler six-sphere $S 6$ is an example of a nearly Kähler manifold that is not Kähler.^[1] The familiar almost complex structure on the six-sphere is not induced by a complex atlas on $S 6$ .

A nearly Kähler manifold should not be confused with an almost Kähler manifold. An almost Kähler manifold $M$ is an almost Hermitian manifold with a closed Kähler form: $d ω = 0$ . The Kähler form or fundamental 2-form $ω$ is defined by

$\omega(X,Y) = g(JX,Y), \,$

where $g$ is the metric on $M$ . The nearly Kähler six-sphere is an example of a nearly Kähler manifold that is not almost Kähler.

References

^ F. Dillen and L. Verstraelen, ed. Handbook of Differential Geometry, volume II. ISBN 978-0444822406. North Holland.

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